The present invention relates to optical manipulation of molecules. More specifically, the present invention relates to a multiple single frequency laser and method of optical manipulation of multi-state-molecules, e.g., cooling or heating of molecules or atoms, and population transfer.
Atomic laser cooling is known and generally involves cooling external degrees of freedom (e.g., translation). However, cooling internal degrees of freedom (e.g., vibration, and rotation), as well as external, is also known, see, A. Kastler, J. Phys. Rad. 11, 255 (1950), wherein the concept of "lumino-refrigeration" is introduced. Laser cooling of internal degrees (i.e., molecules) is discussed in N. Djeu and W. T. Whitney, Phys. Rev. Lett. 46, 236 (1981), wherein significant cooling was measured of CO.sub.2 molecules in a cell with radiation from a CO.sub.2 laser.
Most laser cooling involves external degrees of freedom (i.e., translation), wherein the Doppler effect is typically employed. In general, Doppler cooling (see, W. D. Phillips, Ann. Phys. Fr. 10, 717 (1985)) is used to create a cooling cycle whereby a particle (i.e., an atom or molecule) is made to spontaneously radiate photons of greater energy than those it absorbs from a laser. The cooling cycle has just two steps: (1) absorption (FIG. 1) followed by (2) spontaneous emission (FIG. 2). In the first step (FIG. 1), a particle traveling with some velocity (V) with respect to the laser wave vector (k.sub.L) absorbs a photon. The resonance condition for this absorption can be expressed as: EQU v.sub.Absorption =v.sub.Laser =v.sub.E +k.sub.L .multidot.V=v.sub.E +k.sub.L VCos.theta. (1)
Where v.sub.E is the "rest" velocity resonance frequency of the particle and v.sub.L and k.sub.L are the frequency and wave vector of the laser, respectively, and .theta. is the angle between k.sub.L and V. In the second step (FIG. 2), which occurs on the average one radiative life .tau..sub.rad after the first step, the particle radiates a photon at frequency v.sub.E, which is the characteristic or resonance frequency. This is identical to the frequency of absorption the same particle would have if it had no velocity component parallel k.sub.L. The energy (and momenta) of the absorbed v.sub.A and radiated v.sub.E photons are different. The key to cooling is that the radiated frequency (and therefore energy) in each cycle be greater than that of absorption. The difference in energy in a given cycle may be written as: EQU .DELTA.E=h(v.sub.L -v.sub.E) (2)
Where (h) is Planck's constant. After each cycle the particle velocity is reduced by an amount given by: ##EQU1##
Where v.sub.E is the resonance frequency, M is the mass, and C is the speed of light. The resulting reduction in particle velocity in this cycle, means that the absorption frequency for the next cycle will occur at a higher frequency. The Doppler shift associated with .DELTA. V is given by (see, W. D. Phillips, Ann. Phys. Fr. 10, 717 (1985)): ##EQU2##
The particle changes its Doppler shift as it cools (i.e., v.sub.A and v.sub.L increase as the particle cools, however, v.sub.E does not change). Hence, to cool a particle from some initial velocity to zero, it is necessary to allow for this changing Doppler shift.
The acceleration a particle experiences as it cools is given by: ##EQU3## Where k.sub.L is the wave vector of the laser and .tau..sub.rad is the spontaneous lifetime of the particle. The number of photons required to completely cool the particle is given by: ##EQU4##
Where the (most probable) initial velocity of the particle is given by: ##EQU5##
Where T is the translational temperature and K is the Boltzmann constant. The time required to Doppler cool a particle is determined by the rate limiting step in the cooling cycle, i.e., the spontaneous lifetime of the resonance (.tau..sub.rad). Since most translational cooling is done with electronic transitions that have very short radiative lifetimes (generally, on the order of 10 nanoseconds), the time needed to cool a particle can be quite short (e.g., on the order of milliseconds). Consequently, the minimum distance needed to stop the particle is given by: ##EQU6## which can also be made quite small.
In the case of momentum exchange; because spontaneous emission can occur in any direction, after many cycles in succession the momenta of the radiated photons average an exceedingly small quantum limiting value resulting in a minimum temperature of the particle given by: ##EQU7## Where .DELTA. v is the radiative linewidth of the transition in angular frequency units. With narrow band lasers, the frequency difference between absorbed and emitted photons can be made small enough to realize this limit. As a result, the residual kinetic energy (and therefore the velocity) can be reduced to very small value (e.g., on the order of millimeters per second).
In vibrational cooling (FIG. 3), the molecule in vibrational level (v) first absorbs a photon from the cooling laser, transferring it to an excited electronic state (i.e., absorption). Next, the molecule radiates a photon of higher energy (i.e., emission). The two steps together result in the transfer of the molecule from a higher to a lower vibrational level (v-1). This process is referred to as vibrational electronic cooling. The energy change in vibrational cooling is given approximately by: EQU .DELTA.E.congruent.w.sub.e ( 10)
where w.sub.e is the vibrational constant (in energy units) of the vibrational mode (more precise formula uses the Dunham expansion, see, G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand-Reinhold, New York (1950).
In rotational cooling (FIG. 4) the same scenario results, the cooling cycle consists of absorption of a photon from a laser out of a ground state level (J+1) to an excited electronic state. After approximately one radiative lifetime, the molecule radiates a photon of higher energy, transferring it to a lower (J-1) rotational level. The molecular internal energy is thus reduced by an amount given by: EQU .DELTA.E.congruent.4B(J+1/2) (11)
(see, G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand-Reinhold, New York (1950)). This process is referred to as RE cooling. Subsequent cooling cycles to reduce rotation and vibration further require successively higher frequency (and energy) laser photons, until the lowest rotation and vibration states (J=0 and v=0) are reached. Formuli for these shifts may also be derived from the appropriate Dunham expansions, see, G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand-Reinhold, New York (1950).
TABLE 1, below, shows that the degrees of freedom of a particle may be divided into external (i.e., translation) and internal (i.e., vibration, rotation and electronic). The cooling types that result depend on the type of optical transition that is used for the cooling cycle. Optical transitions are categorized herein as: electronic, vibrational, or rotational. Translational cooling can be performed on electronic, vibrational, or rotational transitions. In the case of internal degrees of freedom there are more possibilities. For vibrational cooling, electronic or vibrational transitions can be used. Rotational cooling can be performed on electronic, vibrational, and rotational transitions. Finally, for electronic degrees of freedom, just electronic transitions can be used.
TABLE 1 ______________________________________ Degree of Freedom Optical Transition Cooling Type ______________________________________ EXTERNAL: Translation Electronic TE Vibration TV Rotation TR INTERNAL: Vibration Electronic VE Vibration VV Rotation Electronic RE Vibration RV Rotation RR Electronic Electronic EE ______________________________________
The rate of cooling in each of the above cases is proportional to the reciprocal of the spontaneous radiative lifetime of the optical transition (or Einstein A coefficient) multiplied by the average energy change in a given cycle. The radiative rates (proportional to the Einstein A coefficient) for optical transitions lie in the order: EQU Electronic&gt;Vibrational&gt;Rotational
Furthermore, the cooling cycle energy difference for each degree of freedom is generally in the order: EQU Electronic&gt;Vibrational&gt;Rotational&gt;Translational
With these approximations, overall ordering of the rates of the cooling types becomes: EQU EE&gt;VE&gt;RE&gt;TE&gt;VV&gt;RV&gt;TV&gt;RR&gt;TR
Heretofore, only TE and RV cooling have been successfully realized experimentally. Cooling can also be achieved by converting translational degrees of freedom into rotational and vibrational using electronic transitions. This process is known as photoassociation and is well known in the art.